Satellite-Galaxy Planes as Coherence-Anchoring Minima

1 Observed facts to explain

• Ultra-thin planes – rms height ≤ 10 kpc at radii 50–250 kpc
  (MW, M31, Cen A).
• Co-rotating membership – ≥ 80 % of satellites share the same
  orbital sense in the plane.
• Hinted spin alignment – dwarf spins tend to lie in the host’s disc plane.

ΛCDM predicts only mild triaxial flattening and randomized orbital poles; the probability of getting MW + M31 style planes in zoom simulations is less than 1 %.

2 PBG physical picture

• The host disc is itself a large-area coherence mode.

• Its envelope generates a scalar coherence potential

  $${\mathrm{\Phi }}_{\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{t}}(R,z)\approx \frac{Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}}{4\pi \alpha R}{e}^{-kR}\exp [-\frac{z^2}{2{\sigma }_{\perp }^2}],$$

  with $k=\sqrt{\beta /\alpha }$ and ${\sigma }_{\perp }$ the disc scale-height.

• A satellite mode of total coherence-mass $M_{\mathrm{c}\mathrm{o}\mathrm{h}}$ minimises the static cost

  $${\mathcal{C}}_{\mathrm{s}\mathrm{a}\mathrm{t}}=M_{\mathrm{c}\mathrm{o}\mathrm{h}}{\mathrm{\Phi }}_{\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{t}}(R,z).$$

• Force (correct sign)

  $$\begin{array}{}\text{(2.1)}& \ddot{\mathbf{r}}=-\mathrm{\nabla }{\mathrm{\Phi }}_{\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{t}}(R,z).\end{array}$$

  In particular the vertical component is

  $$\ddot{z}=-\frac{A}{R{\sigma }_{\perp }^2}z{e}^{-kR}{e}^{-z^2/2{\sigma }_{\perp }^2}\stackrel{|z|\ll {\sigma }_{\perp }}{⟶}-{\omega }_z^{2}z,{\omega }_z^2\equiv \frac{A{e}^{-kR}}{R{\sigma }_{\perp }^2},$$

  where $A\equiv Q_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}/(4\pi \alpha )$.

3 Settling time → plane thickness

For small $|z|$ the equation is harmonic with damping by orbital mixing; solution

Numerical estimate (Milky-Way like host)

Plugging gives

So an initially 200 kpc-thick cloud collapses to

matching observed planes without fine-tuning.

4 Co-rotation and spin alignment

• Phase continuity: a satellite that flips its orbital direction introduces a $2\pi$ discontinuity in $\varphi$ across the disc plane ⇒ extra anchoring cost; prograde paths are favoured.
• Torque on internal spin: integrating the cross-force$${\tau }_z\propto \int (\mathbf{r}\times \mathrm{\nabla }{\mathrm{\Phi }}_{\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{t}})d^{3}x$$shows ${\dot{\psi }}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}\sim -{\tau }_z/I$, giving alignment times comparable to $\tau$ for dwarfs (low $I$).

5 Prediction table

Upcoming deep proper-motion surveys (Roman, LSST) can test the $H(R)\propto R^{1/2}$ law and the > 80 % prograde fraction out to 250 kpc; ΛCDM has no parameter-free way to mimic both simultaneously.

6 Take-away

In PBG the host disc’s coherence mode actively corrals satellites into razor-thin, co-rotating sheets; the same α, β constants fixed by lensing and Lamb shift set the collapse rate—no dark-matter sub-halo gymnastics or rare filamentary accidents required.
Satellite planes therefore provide a sharp astrophysical falsifier: if future all-sky surveys find isotropic, counter-rotating dwarf populations, the coherence-anchoring picture fails.

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